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The new shape of music: Music has its own geometry, researchers find
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And the so-called musica universalis or "music of the spheres" emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies -- the sun, moon and planets -- could be viewed as a form of music, inaudible but perfectly harmonious.
Now, three music professors – Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University -- have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.
The trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families." They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.
Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.
The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph's University in Philadelphia. In an accompanying essay, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content."
The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like "chord" are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.
"The music of the spheres isn't really a metaphor -- some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. "The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before."
Like what?
"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualization tools -- imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas."
"But to me," Tymoczko added, "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries."
Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle "C" on a piano, followed by the note "E" above that and the note "G" above that. Musicians have many different terms to describe this sequence of events, such as "an ascending C major arpeggio," "a C major chord," or "a major chord." The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.
The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the "OPTIC symmetries," with each letter of the word "OPTIC" representing a different way of ignoring musical information -- for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.
In this way, the musicians are able to reduce musical works to their mathematical essence.
Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.
The method could help answer whether there are new scales and chords that exist but have yet to be discovered.
"Have Western composers already discovered the essential and most important musical objects?" Tymoczko asked. "If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization."
The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.
Source: Princeton University
The figure shows how geometrical music theory represents four-note chord-types -- the collections of notes form a tetrahedron, with the colors indicating the spacing between the individual notes in a sequence. In the blue spheres, the notes are clustered, in the warmer colors, they are farther apart. The red ball at the top of the pyramid is the diminished seventh chord, a popular 19th-century chord. Near it are all the most familiar chords of Western music.The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.
And the so-called musica universalis or "music of the spheres" emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies -- the sun, moon and planets -- could be viewed as a form of music, inaudible but perfectly harmonious.
Now, three music professors – Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University -- have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.
The trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families." They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.
Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.
The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph's University in Philadelphia. In an accompanying essay, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content."
The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like "chord" are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.
"The music of the spheres isn't really a metaphor -- some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. "The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before."
Like what?
"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualization tools -- imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas."
"But to me," Tymoczko added, "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries."
Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle "C" on a piano, followed by the note "E" above that and the note "G" above that. Musicians have many different terms to describe this sequence of events, such as "an ascending C major arpeggio," "a C major chord," or "a major chord." The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.
The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the "OPTIC symmetries," with each letter of the word "OPTIC" representing a different way of ignoring musical information -- for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.
In this way, the musicians are able to reduce musical works to their mathematical essence.
Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.
The method could help answer whether there are new scales and chords that exist but have yet to be discovered.
"Have Western composers already discovered the essential and most important musical objects?" Tymoczko asked. "If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization."
The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.
Source: Princeton University
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friendster7Re: Ultimate NEWS
Graphene used to create world's smallest transistor
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A Manchester researcher shows graphene quantum dots on a chip.
Researchers have used the world's thinnest material to create the world's smallest transistor, one atom thick and ten atoms wide.
Reporting their peer-reviewed findings in the latest issue of the journal Science, Dr Kostya Novoselov and Professor Andre Geim from The School of Physics and Astronomy at The University of Manchester show that graphene can be carved into tiny electronic circuits with individual transistors having a size not much larger than that of a molecule.
The smaller the size of their transistors the better they perform, say the Manchester researchers.
In recent decades, manufacturers have crammed more and more components onto integrated circuits. As a result, the number of transistors and the power of these circuits have roughly doubled every two years. This has become known as Moore's Law.
But the speed of cramming is now noticeably decreasing, and further miniaturisation of electronics is to experience its most fundamental challenge in the next 10 to 20 years, according to the semiconductor industry roadmap.
At the heart of the problem is the poor stability of materials if shaped in elements smaller than 10 nanometres in size. At this spatial scale, all semiconductors -- including silicon -- oxidise, decompose and uncontrollably migrate along surfaces like water droplets on a hot plate.
Four years ago, Geim and his colleagues discovered graphene, the first known one-atom-thick material which can be viewed as a plane of atoms pulled out from graphite. Graphene has rapidly become the hottest topic in physics and materials science.
Now the Manchester team has shown that it is possible to carve out nanometre-scale transistors from a single graphene crystal. Unlike all other known materials, graphene remains highly stable and conductive even when it is cut into devices one nanometre wide.
Graphene transistors start showing advantages and good performance at sizes below 10 nanometres - the miniaturization limit at which the Silicon technology is predicted to fail.
"Previously, researchers tried to use large molecules as individual transistors to create a new kind of electronic circuits. It is like a bit of chemistry added to computer engineering", says Novoselov. "Now one can think of designer molecules acting as transistors connected into designer computer architecture on the basis of the same material (graphene), and use the same fabrication approach that is currently used by semiconductor industry".
"It is too early to promise graphene supercomputers," adds Geim. "In our work, we relied on chance when making such small transistors. Unfortunately, no existing technology allows the cutting materials with true nanometre precision. But this is exactly the same challenge that all post-silicon electronics has to face. At least we now have a material that can meet such a challenge."
"Graphene is an exciting new material with unusual properties that are promising for nanoelectronics", comments Bob Westervelt, professor at Harvard University. "The future should be very interesting".
Source: University of Manchester -
friendster7Re: Ultimate NEWS
Nanotechnology paves way for super iPods
A breakthrough by scientists from the University of Glasgow could see the storage capacity of an iPod increase 150,000 times.
Nanotechnology researchers have developed a molecule-sized switch which means that data storage can be dramatically increased without the need to increase the size of devices.
Professor Lee Cronin and Dr Malcolm Kadodwala’s work would see 500,000 gigabytes squeezed onto one square inch. The current limit for the space is around 3.3 gigabytes.
The researchers believe that their development could see the number of transistors per chip rising from today’s limit of 200million to well over one billion.
Professor Lee Cronin said: “What we have done is find a way to potentially increase the data storage capabilities in a radical way.
“We have been able to assemble a functional nanocluster that incorporates two electron donating groups, and position them precisely 0.32 nm apart so that they can form a totally new type of molecular switching device.
“This is unprecedented and provides a route to produce new a molecule-based switch that can be easily manipulated using an electric field.
“By taking these nano-scale clusters, just a nanometer in size, and placing them onto a gold or carbon, we can control the switching ability. Not only is this a new type of switchable molecule, but by grafting the molecule on to metal (gold) or carbon means that we can potentially bridge the gap between traditional semiconductor devices and components for nanoscale plastic electronics.
“The key advantage of the molecule sized switch is information / transistor density in traditional semi-conductors. Molecule sized switches would lead to increasing data storage to say 4 Petabits per square inch.
“This breakthrough shows conceptually that this is possible (showing the bulk effect) but we are yet to solve the fabrication and addressing problems.
“The fact these switches work on carbon means that they could be embedded in plastic chips so silicon is not needed and the system becomes much more flexible both physically and technologically.
“Since these switches are little balls of metal oxide they are made of similar stuff to normal semi-conductors but are much easier to manipulate as discrete molecular units.
Source: University of Glasgow -
friendster7Re: Ultimate NEWS
Scientists Show First 3-D Image of Antibody Gene
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The 3-D structure of the immunoglobulin locus in B cells is shown, with the relative positions of the different portions of the immunoglobulin genes. Grey objects indicate constant regions. Blue objects indicate proximal variable regions. Green objects indicate distal variable regions. Red line indicates the linker connecting the proximal variable and joining regions.
Using a multidisciplinary mix of geometry, biological research and techniques developed to solve problems on supercomputers, scientists at the University of California, San Diego have shown for the first time how a genome is organized in three-dimensional space.
Researchers led by Cornelis Murre, a professor of biology at UC San Diego, and Steve Cutchin, senior scientist for visualization services at the San Diego Supercomputer Center (SDSC), used the gene encoding the immunoglobulin heavy chain locus — responsible for generating diverse kinds of antibodies — to demonstrate the structure of the genome.
The observations, the researchers say, permit an insight into the structure of the human genome, which until now has remained elusive.
Because the genome is the most essential part of the cell for storing and accessing genetic information, the complete DNA sequence of a wide variety of genomes has been revealed in studies performed in a large number of laboratories — “a tremendous success that has provided insight into mechanisms that underpin the development of a wide variety of diseases,” the authors say.
However, Murre said, “it has remained unclear as to how the genome is organized in three-dimensional space. This is an important issue since the regulation of gene expression is controlled by interactions of genomic elements that are separated by large genomic distances. Thus, our team wanted to determine how the genome is structured within the nucleus.”
The experiments described in the Cell paper, he said, provide a first glimpse into this question. “As a model system, we used the gene encoding for the immunoglobulin heavy chain locus, because it is responsible for generating the wide diversity of antibodies.”
Having measured the distances that separate the various parts of the gene, Murre said, the researchers, in collaboration with Cutchin at the SDSC, then used geometry to resolve the first structure of a genetic locus.
His work, said Cutchin, involved computational geometry, scientific visualization, computational methods and numerical methods.
“The resulting structure shows that the antibody gene is organized into ‘flower-like’ structures that are connected by linkers,” said Murre. “These flowers contain the various parts that ultimately generate the wide variety of antibodies. This is the first time that geometry has been used to determine the structure of a genetic locus. Ultimately, the same approach should be used to elucidate the structure of the entire human genome.”
Contributing equally to the work were Suchit Jhunjhunwala, Mandy M. Peak, and Menno C. van Zelm, all with the Division of Biological Sciences at UC San Diego; Roy Riblet of the Torrey Pines Institute for Molecular Studies; Jacques J.M. van Dongen and Frank G. Grosveld of Erasmus MC in Rotterdam, The Netherlands; and Tobias A. Knoch of Heidelberg University, Germany.
Source: UCSD
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